WASHINGTON—President Donald Trump’s new national-security adviser has suddenly become the administration’s political lightning rod in talks with North Korea, which delivered an unusually personal attack on Wednesday in threatening to pull out of June’s high stakes summit with the U.S.

John Bolton, known for his bushy white mustache and his confrontational foreign-policy views, is a longstanding figure in the capital’s foreign-policy circles and has quickly established himself as the president’s prime confidant since taking...